Question

2logx + log 1/3-log 3/4 =2 ​

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:x\:=\:\dfrac{3b}{2}}}}$

Step-by-step-explanation:

The given logarithmic equation is

$\displaystyle{\sf\:2\:\log\:(\:x\:)\:+\:\log\:\left(\:\dfrac{1}{3}\:\right)\:-\:\log\:\left(\:\dfrac{3}{4}\:\right)\:=\:2}$

We have to find the value of x.

Now,

$\displaystyle{\sf\:2\:\log\:(\:x\:)\:+\:\log\:\left(\:\dfrac{1}{3}\:\right)\:-\:\log\:\left(\:\dfrac{3}{4}\:\right)\:=\:2}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:\left(\:\dfrac{x}{y}\:\right)\:=\:\log_b\:(\:x\:)\:-\:\log_b\:(\:y\:)}}$

$\displaystyle{\implies\sf\:2\:\log\:(\:x\:)\:+\:\log\:\left(\:\dfrac{\dfrac{1}{3}}{\dfrac{3}{4}}\:\right)\:=\:2}$

$\displaystyle{\implies\sf\:2\:\log\:(\:x\:)\:+\:\log\:\left(\:\dfrac{1}{3}\:\times\:\dfrac{4}{3}\:\right)\:=\:2}$

$\displaystyle{\implies\sf\:2\:\log\:(\:x\:)\:+\:\log\:\left(\:\dfrac{4}{9}\:\right)\:=\:2}$

$\displaystyle{\implies\sf\:2\:\log\:(\:x\:)\:+\:\log\:\left[\:\left(\:\dfrac{2}{3}\:\right)^2\:\right]\:=\:2}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:(\:a^k\:)\:=\:k\:\log_b\:(\:a\:)}}$

$\displaystyle{\implies\sf\:2\:\log\:(\:x\:)\:+\:2\:\log\:\left(\:\dfrac{2}{3}\:\right)\:=\:2}$

$\displaystyle{\implies\sf\:2\:\left[\:\log\:(\:x\:)\:+\:\log\:\left(\:\dfrac{2}{3}\:\right)\:\right] \:=\:2}$

$\displaystyle{\implies\sf\:\log\:(\:x\:)\:+\:\log\:\left(\:\dfrac{2}{3}\:\right)\:=\:1}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:(\:x\:)\:+\:\log_b\:(\:y\:)\:=\:\log_b\:(\:xy\:)}}$

$\displaystyle{\implies\sf\:\log\:\left(\:\dfrac{2x}{3}\:\right)\:=\:1}$

We know that,

$\displaystyle{\pink{\sf\:\log_b\:(\:b\:)\:=\:1}}$

$\displaystyle{\implies\sf\:\log_b\:\left(\:\dfrac{2x}{3}\:\right)\:=\:\log_b\:(\:b\:)}$

Taking antilog to the base b on both sides, we get,

$\displaystyle{\implies\sf\:\dfrac{2x}{3}\:=\:b}$

$\displaystyle{\implies\sf\:2x\:=\:3b}$

$\displaystyle{\implies\underline{\boxed{\red{\sf\:x\:=\:\dfrac{3b}{2}}}}}$

Answer 2

$\huge\mathfrak\red{answer}$

$x = \frac{3b}{2}$